Structural properties of 2-C-normal operators
Messaoud Guesba, Ismail Lakehal, Sid Ahmed Ould Ahmed Mahmoud
TL;DR
This paper investigates a new operator class, the $2$-$C$-normal operators, on complex Hilbert spaces, enriching complex symmetric operator theory by incorporating a conjugation. It defines $2$-$C$-normalness by the condition $(SC)^2 (SC)^{\#}=(SC)^{\#}(SC)^2$, and derives foundational properties, including scalar stability, invertibility preservation, and unitary invariance. Key contributions include explicit examples of $2$-$C$-normal operators that are not $C$-normal, the result that $(SC)^2$ is normal when $S$ is $2$-$C$-normal, and the norm-closedness of the class $N_{2.C}(\ K)$, along with several relations to $2$-normality via $CSC=S$. These findings deepen understanding of conjugation-influenced operator behavior and lay groundwork for further study in complex symmetric operator theory.
Abstract
In this paper, we introduce and study a new class of bounded linear operators on complex Hilbert spaces, which we call 2-C-normal operators. This class is inspired by and closely related to the notion of 2-normal operators, with additional structure imposed via conjugation. We investigate various fundamental properties of 2-C-normal operators, including algebraic characterizations, spectral properties, and examples that distinguish them from classical normal and 2-normal operators. Additionally, we explore how this class behaves under common operator transformations and examine its relation to other well-studied families of operators. The results presented contribute to a deeper understanding of conjugation-influenced operator behavior and open avenues for further study in the context of complex symmetric operator theory.